2021/11/10 by Timothy C. Burness, Burness, Timothy C., Andrea Lucchini +3 · 1 citation
Mathematics · Neuroscience · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Nuclear Receptors and Signaling
paper · pdf · doi:10.48550/arxiv.2111.05697
openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G ∖ R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS(G), is at most 5. More precisely, we show that δS(G) \leqslant 3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS(Sn) = 3 for all n \geqslant 6. We also establish the existence of simple groups with δS(G) \geqslant 4. For instance, we prove that δS(A2p+1) \geqslant 4 for every Sophie Germain prime p \geqslant 5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.